To determine which of Jupiter's moons has the greatest gravitational force with Jupiter, we need to use the formula for gravitational force:
[tex]\[ F = G \frac{m_1 m_2}{d^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of Jupiter, [tex]\( 1.898 \times 10^{27} \, \text{kg} \)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the moon,
- [tex]\( d \)[/tex] is the distance between the moon and Jupiter (converted to meters).
Given the mass of the moons and their respective distances from Jupiter, we can calculate the gravitational force for each moon.
### Io
- Mass: [tex]\( 8.932 \times 10^{19} \)[/tex] kg
- Distance: [tex]\( 421,700 \)[/tex] km [tex]\( = 421,700,000 \)[/tex] m
### Europa
- Mass: [tex]\( 4.8 \times 10^{19} \)[/tex] kg
- Distance: [tex]\( 671,034 \)[/tex] km [tex]\( = 671,034,000 \)[/tex] m
### Ganymede
- Mass: [tex]\( 14.819 \times 10^{19} \)[/tex] kg
- Distance: [tex]\( 1,070,412 \)[/tex] km [tex]\( = 1,070,412,000 \)[/tex] m
### Callisto
- Mass: [tex]\( 10.759 \times 10^{19} \)[/tex] kg
- Distance: [tex]\( 1,882,709 \)[/tex] km [tex]\( = 1,882,709,000 \)[/tex] m
The gravitational forces calculated for each moon are:
- Io: [tex]\( 6.36272926176099 \times 10^{19} \)[/tex] N
- Europa: [tex]\( 1.350374156878066 \times 10^{19} \)[/tex] N
- Ganymede: [tex]\( 1.6383960469311234 \times 10^{19} \)[/tex] N
- Callisto: [tex]\( 3.8450982545003044 \times 10^{18} \)[/tex] N
Comparing these forces, Io has the greatest gravitational force with Jupiter at [tex]\( 6.36272926176099 \times 10^{19} \)[/tex] N.
Hence, the moon with the greatest gravitational force with Jupiter is Io.
